Problem 84 - Algebra for College Students

Question

Find each product. Assume that the variables in the exponents represent positive integers. For example, $$ \left(x^{2 n}\right)\left(x^{3 n}\right)=x^{2 n+3 n}=x^{5 n} $$ $$\left(4 x^{2 n-1}\right)\left(-3 x^{n+1}\right)$$

Step-by-Step Solution

Answer

The product is \(-12x^{3n}\).

Step 1: Identify the components of each term

We have two terms: \(4x^{2n-1}\) and \(-3x^{n+1}\). Each term has a coefficient (\(4\) and \(-3\) respectively) and a variable component \(x\) with exponents \(2n-1\) and \(n+1\).

Step 2: Multiply the coefficients

Multiply the coefficients from each term. \(4\) and \(-3\) give us \(4 \times -3 = -12\).

Step 3: Apply the product of powers property

For the variable part, we use the property \(x^a \cdot x^b = x^{a+b}\). The exponents are \(2n-1\) and \(n+1\). So, we add these exponents: \[(2n-1) + (n+1) = 2n + n - 1 + 1 = 3n\]

Step 4: Write the final expression

Combine the coefficient and the variable with the combined exponent to form the final product: \(-12x^{3n}\).

Key Concepts

Exponential Expressions

Exponential expressions are a fundamental part of algebra. They involve numbers or variables raised to a power, denoted as exponents. For instance, in the expression \(x^n\), the number \(x\) is the base, and \(n\) is the exponent. Exponents indicate how many times a base is multiplied by itself.

Understanding exponential expressions is crucial because they help to simplify the representation of large numbers and facilitate operations involving repeated multiplication. This simplification becomes particularly handy when working with algebraic expressions where you perform operations on variables with exponents.

When dealing with exponential expressions, one must pay attention to fundamental properties such as the product of powers property. This allows for the simple multiplication of like bases.

Algebraic Expressions

Algebraic expressions are combinations of variables, constants, and arithmetic operators such as addition, subtraction, multiplication, and division. They represent a wide range of mathematical relationships and can be as simple as \(x + 2\) or as complex as \(4x^{2n-1}\).

In algebraic expressions, knowing how to organize and manipulate terms is vital. Terms in an expression are separated by addition or subtraction operators. A term can be a constant, like \(4\), a variable, like \(x\), or a product like \(4x\), which involves a constant (coefficient) and a variable.

When simplifying or expanding algebraic expressions, especially those with exponents, remember the rules governing operations such as multiplication and the distributive property. Breaking down each term into its components (coefficients and exponentiated variables) helps clarify the process and avoid potential errors.

Product of Powers Property

The product of powers property is a key tool in simplifying products of exponential expressions with the same base. It states that for any base \(x\), and any exponents \(a\) and \(b\), \(x^a \cdot x^b = x^{a+b}\).

This principle is straightforward yet powerful. It allows the combination of terms simply by adding their exponents. This is possible because multiplying terms with the same base is akin to extending the series of repeated multiplications. Thus, instead of calculating each power separately, they can be combined by summing the exponents.

To use this property effectively, first ensure you have terms with the same base. Then, simply add their exponents to find the new expression's exponent. This simplification step is essential in more complicated calculations, aiding in reducing time and effort.